Algebraic Boundary Research Articles

Convex hulls of surfaces in fourspace

AbstractThis is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

Combinatorics of Correlated Equilibria

We study the correlated equilibrium polytope PG of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of PG , and show that for ( 2 × n ) -games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic ( 2 × 3 ) -games.

Inter relations between algebraic boundary and other boundary

Inter relations between algebraic boundary and other boundary

Triangular Bernstein moment-based identification of algebraic curves

Extending our previous results, in this paper we present a theoretical improvement of a strategy for the identification of binary images with algebraic boundaries. Such identification is obtained from few samples and it is based on a representation of the image shape in terms of non-separable bivariate Bernstein polynomials piecewisely defined over triangular domains.

Minimal boundaries for operator algebras

We study boundaries for unital operator algebras. These are sets of irreducible ∗ * -representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of C ∗ \mathrm {C}^* -algebras, our techniques allow us to provide a new proof of a recent characterization of those C ∗ \mathrm {C}^* -algebras admitting only finite-dimensional irreducible representations.

The Algebraic Boundary of the Sonc-Cone

In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semialgebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we give sufficient conditions for equality between the sonc-cone and the sparse nonnegativity cone for generic support sets, and we describe a semialgebraic stratification of the boundary of the sonc-cone in the univariate case.

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.

Intersection bodies of polytopes

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.

Linear differential-algebraic boundary value problem with singular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M. O. Bogolybov, A. D. Myshkis, A. M. Samoilenko, M. O. Perestyk and O. A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action generalizes the results of S. Campbell, A. M. Samoilenko, M. O. Perestyuk and O. A. Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

Linear differential-algebraic boundary value problem with nonsingular pulse influence

The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N.N. Luzin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of pulse boundary value problems for differential equations, initiated M.O. Bogolybov, A.D. Myshkis, A.M. Samoilenko, M.O. Perestyk and O.A. Boichuk. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk on a pulse linear boundary value problems for differential-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a pulse linear boundary value problem for a differential-algebraic equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a differential-algebraic equation with nonsingular impulse action. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a differential-algebraic equation with impulse action in the critical case in this article can be transferred to the linear differential-algebraic boundary value problem for a differential-algebraic equation with singular impulse action. The above scheme of the analysis of the seminonlinear differential-algebraic boundary value problems with impulse action neralizes the results of S. Campbell, A.M. Samoilenko, M.O. Perestyuk and O.A Boichuk and can be used for proving the solvability and constructing solutions of weakly nonlinear boundary value problems with singular impulse action in the critical and noncritical cases.

ПРО ПОРУШЕННЯ ЄДИНОСТІ РОЗВ’ЯЗКУ ЗАДАЧІ ДІРІХЛЕ ДЛЯ СИСТЕМ ДРУГОГО ПОРЯДКУ

The study of the issues of the correct posedness of boundary value problems for differential equations and systems occupies an important place in modern research. When considering correctness, the question of unique solvability of this problem is of paramount importance. In particular, the problem of violation of the uniqueness of the solution of boundary value problems for general differential equations in bounded domains with algebraic boundary is of interest. The property of nontrivial solvability of the homogeneous Dirichlet problem for incorrectly elliptic equations of the second order was first pointed out by A. V. Bitsadze, having constructed an example of an equation with constant complex obtained a condition for the violation of the uniqueness of the solution to the Dirichlet problem in the unit disc for a hyperbolic equation in the case when the slope angles of the characteristics differ in sign. V. P. Burskii, considering the homogeneous Dirichlet problem in the unit disc for second-order equations with constant complex coefficients and a homogeneous non-degenerate symbol, obtained a criterion for nontrivial solvability in the form of π-irrationality of the angle between the characteristics. In this paper, we investigate the question of violation of the uniqueness of the solution of the homogeneous Dirichlet problem for a system of typeless second-order partial differential equations in a model domain – a circle. The original system is written in the form of an equation with commuting matrix coefficients. The permutability condition allows one to obtain a necessary and sufficient condition for the nontrivial solvability of the problem under consideration in the form of equality to zero of the determinant, the elements of which are expressed in terms of the coefficients of the equation. This form of writing the criterion allows one to construct examples of systems for which the kernel of the Dirichlet problem is nontrivial and infinite-dimensional. The study was based on the integral condition for the connection of associated boundary L-traces, as well as a functional scheme, the application of which reduces the expansion of a matrix function in a Fourier series to a standard expansion of each of its elements. A theorem of nontrivial solvability of the homogeneous Dirichlet problem is proved.

Differential algebraic fast multipole-accelerated boundary element method for nonlinear beam dynamics in arbitrary enclosures

A novel method is developed to take into account realistic boundary conditions in intense nonlinear beam dynamics. The algorithm consists of three main ingredients: the boundary element method that provides a solution for the discretized reformulation of the Poisson equation as boundary integrals; a novel fast multipole method developed for accurate and efficient computation of Coulomb potentials and forces; and differential algebraic methods, which form the numerical structures that enable and hold together the different components. The fast multipole method, without any modifications, also accelerates the solution of intertwining linear systems of equations for further efficiency enhancements. The resulting algorithm scales linearly with the number of particles $N$, as $m\text{ }\mathrm{log}\text{ }m$ with the number of boundary elements $m$, and, therefore, establishes an accurate and efficient method for intense beam dynamics simulations in arbitrary enclosures. Its performance is illustrated with three different cases and structures of practical interest.

Stable recovery of planar regions with algebraic boundaries in Bernstein form

We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available.

Local Energy Estimates for the Fractional Laplacian

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singula...

Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree

We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Voronoi cells of varieties

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells.

Rigidity Theorem by the Minimal Point of the Bergman Kernel

We use the Suita conjecture (now a theorem) to prove that for any domain $\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ for some $z_0 \in \Omega$ if and only if $\Omega$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) closed polar set. When $\Omega$ is bounded with $C^{\infty}$-boundary, we provide a simple proof of this using the zero set of the Szeg\"o kernel. Finally, we show that this theorem fails to hold in $\mathbb{C}^n$ for $n > 1$ by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$.

Roller boundaries for median spaces and algebras

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebras, where we recover the zero-completions of Bandelt and Meletiou. Along the way, we prove various properties of halfspaces in finite rank median spaces and a duality result for locally convex median spaces.

Generating (co)homological information using boundary scale

In this paper we develop a new computational technique called boundary scale-space theory. This technique is based on the topol11comples -> complexogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects {Ki}i ≥ 1 all of them having the same number of closed pieces than K. Each piece of Ki (∀i ≥ 1) presents the same interior part than the corresponding one in K, and a different boundary part depending on the scale i. Working with coefficients in a field, a scale is installed for the algebraic boundary of each piece and a new invariant for cell complex isomorphisms is given in terms of the Betti numbers of the generated boundary-scale-space cell complexes. Moreover, the so called homology boundary scale-space model of K (hbss-model for short) is introduced here. This model consists of a hierarchical graph whose nodes are the homology generators of the different boundary scale levels and whose edges are specified by homology generators of consecutive boundary scale indices linked by (hbss-transition maps) preserving homology classes. Various codes for each connected subgraph of an hbss-model are defined, which besides being fast and efficient similarity measures for cellular structures, they are as well relevant interpretive tools for the hbss-model. Finally, experimentations mainly aimed at clarifying and understanding the notion of hbss-model, as well as conjecturing about new graph isomorphism invariants (seeing graphs as a 1-dimensional cell complexes), are performed.